Casualty Reinsurance Seminar, June 7th, 2004, Boston
June 7, 2004
““Cat Bond Pricing Using Probability Cat Bond Pricing Using Probability
TransformsTransforms””
published in published in Geneva Papers, 2004Geneva Papers, 2004
Shaun Wang, Ph.D., FCAS
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Shaun Wang, June 2004
What is CAT bond?
A high-yield debt instrument: if the issuer (insurance company) suffers a loss from a particular predefined catastrophe, then the issuer's obligation to pay interest and/or repay the principal is either deferred or forgiven.
Covered events: CA Earthquake, Japan Earthquake, FL Hurricane, EU Winter Storm; Multi-Peril & Multi-territory
Actual-dollar trigger or Reference-index trigger
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Shaun Wang, June 2004
Why CAT bond?
For bond issuers:
Alternative source of capital/capacity for insurance companies with large risk transfer needs
Not subject to the risk of non-collectible reinsurance
For investors:
High yield coupon rate
CAT bond performance is not closely correlated with the stock market or economic conditions.
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Shaun Wang, June 2004
Example of Cat-bond transactions(Data Source: Lane Financial LLC)
Cat bond Transaction
Probability of First $
Loss
Probability of Last $
Loss
Expected Loss given
default
Yields Spread Over
LIBOR
Atlas Re A 0.0019 0.0005 0.5789 2.74%
Atlas Re B 0.0029 0.0019 0.7931 3.75%Atlas Re C 0.0547 0.019 0.5923 14.19%
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Shaun Wang, June 2004
State of the Cat-bond Market
In the past, unfamiliar class of assets to investors, led to limited number of transactions
Phenomenal performance of CAT bond portfolios, led to recent surge of interest by institutional investors
Cat Bond Market Grew 42% in 2003 Total bond issuance $1.73 billion Reduced cost of issuing (coupon interest and
transaction costs)
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Shaun Wang, June 2004
Cat-bond offers a laboratory for reconciliation of pricing approaches
Capital market pricing is forward-looking:
prices incorporate all available information
No-arbitrage pricing (Black-Scholes Theory)
Actuarial pricing is back-forward looking
Using historical data to project future costs
Explicit adjustments for risk
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Shaun Wang, June 2004
Financial World
Black-Schole-Merton theory for pricing options and corporate credit default risks
A common measure for fund performance is the Sharpe ratio: ={ E[R] r }/[R], the excess return per unit of volatility
also called “market price of risk”
How can we relate it to actuarial pricing?
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Shaun Wang, June 2004
Ground-up Loss X has loss exceedence curve:
SX(t) =1 FX(t) = Pr{ X>t }.
Layer X(a, a+h); a=retention; h=limit
Actuarial World
0
)(][ dttSXE X
dttShaaXEha
a X )()],([
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Shaun Wang, June 2004
Loss Exceedence Curve
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Shaun Wang, June 2004
Insurance prices by layer implies a
transformed distribution– layer (t, t+dt) loss: Slayer (t, t+dt) loss: SXX(t) dt (t) dt
– layer (t, t+dt) price: Slayer (t, t+dt) price: SXX*(t) dt*(t) dt
– implied transform: Simplied transform: SXX(t) (t) S SXX*(t)*(t)
Venter 1991 ASTIN Paper
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Shaun Wang, June 2004
Insight of Gary Venter (91 ASTIN ):
“Insurance prices by layer imply a transformed distribution”
S(x)=1F(x), or Loss Exceedence Curve
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Shaun Wang, June 2004
Attempt #1 by Morton Lane
(Hachemeister Prize Paper) Morton Lane (2001) “Pricing of Risk Transfer
transactions” proposed a 3-parameter model:
EER = 0.55 (PFL)0.49 (CEL)0.57
PFL: Probability of First Loss
CEL: Conditional Expected Loss (as % of principal)
EER: Expected Excess Return (over LIBOR)
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Shaun Wang, June 2004
Attempt #2: Wang Transform(Sharing 2004 Ferguson Prize with Venter)
Let be standard normal distribution:
(1.645)=0.05, (0)=0.5, (1.645)=0.95
Wang introduces a new transform:
F(x)=0.95, =0.3, F*(x) = 1(1.6450.3) =0.91
Fair Price is derived from the expected value under
the transformed distribution F*(x).
λF(x)F*(x) )(ΦΦ 1
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Shaun Wang, June 2004
WT extends the Sharpe Ratio Concept
If FX is normal(), FX* is normal(+ ):
E*[X] = E[X] + [X]
If FX is lognormal( ), FX* is lognormal(+ )
The transform recovers CAPM & Black-Scholes (ref. Wang, JRI 2000)
extends the Sharpe ratio to skewed distributions
))(()(* 1 xFxF
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Shaun Wang, June 2004
1-factor Wang transformlambda=0.3
-
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
1 11 21 31 41 51 61 71 81 91
Uniform Distribution
Adj
uste
d D
ensi
ty
f(x)f*(x)
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Shaun Wang, June 2004
Unified Treatment of Asset / Loss
The gain X for one party is the loss for the counter party: Y = X
We should use opposite signs of , and we get the same price for both sides of the transaction
))(()( 1* xFxF XX
))(()(
))(()(1*
1*
ySyS
yFyF
YY
YY
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Shaun Wang, June 2004
Baseline Sampling Theory
We have m observations from normal(,2). Not
knowing the true parameters, we have to estimate
and by sample mean & variance.
When assessing the probability of future outcomes,
we effectively need to use Student-t with k=m-2
degrees-of-freedom.
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Shaun Wang, June 2004
Adjust for Parameter Uncertainty
Baseline: For normal distributions, Student-t
properly reflects the parameter uncertainty
Generalization: For arbitrary F(x), we propose the
following adjustment for parameter uncertainty:
))(()( 1* xFQxF
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Shaun Wang, June 2004
A Two-Factor Model
Wang transform with adjustment for parameter uncertainty:
))(()(* 1 yFQyF
where is standard normal CDF, and Q is Student-t CDF with k degrees-of-freedom
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Shaun Wang, June 2004
Student-t Adjustmentk=7
-
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
1 11 21 31 41 51 61 71 81 91
Uniform Distribution
Ad
jus
ted
De
ns
ity
f(x)f*(x)
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Shaun Wang, June 2004
1-factor Wang transformlambda=0.3
-
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
1 11 21 31 41 51 61 71 81 91
Uniform Distribution
Adj
uste
d D
ensi
ty
f(x)f*(x)
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Shaun Wang, June 2004
2-factor transformlambda=0.3, k=7
-
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
1 11 21 31 41 51 61 71 81 91
Uniform Distribution
Ad
jus
ted
De
ns
ity
f(x)f*(x)
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Shaun Wang, June 2004
Insights for the second factor
Explains investor behavior: greed and fear
Investors desire large gains (internet lottery)
Investors fear large losses (market crash)
Consistent with “volatility smile” in option prices
Quantifies increased parameter uncertainty in the tails
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Shaun Wang, June 2004
Empirical Studies
16 CAT-bond transactions in 1999
Fit well to the 2-factor Wang transform
Better fit than Morton Lane’s 3-parameter model (in his
2001 Hachmeister Prize Paper)
12 CAT bond transactions in 2000
Use 1999 estimated parameters to price 2000
transactions, remain to be the best-fit
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Shaun Wang, June 2004
1999 Cat-bond transactions(Data Source: Lane Financial LLC)
Cat bond Transaction
Probability of First $ Loss
Probability of Last $
Loss
Expected Loss given
defaultModel Yields
Spread
Empirical Yields
Spread
Mosaic 2A 0.0115 0.0012 0.3652 3.88% 4.06%Mosaic 2B 0.0525 0.0115 0.541 10.15% 8.36%Halyard Re 0.0084 0.0045 0.75 4.82% 4.56%Domestic Re 0.0058 0.0044 0.8621 4.36% 3.74%Concentric Re 0.0062 0.0022 0.677 4.01% 3.14%Juno Re 0.006 0.0033 0.75 4.15% 4.26%Residential Re 0.0076 0.0026 0.5789 4.08% 3.71%
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Shaun Wang, June 2004
Fit Wang transform to 1999 Cat bondsDate Sources: Lane Financial LLC Publications
Yield Spread for Insurance-Linked Securities
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
Transactions
Yie
ld S
prea
d
Model-Spread
Empirical-Spread
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Shaun Wang, June 2004
Use 1999 parameters to price 2000 Cat Bonds
Fitted versus Empirical Spread
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
Transactions
Yie
ld S
pre
ad
Model-Spread
Empirical-Spread
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Shaun Wang, June 2004
Corporate Bond Default:Historial versus Implied Default Frequency
Corporate Bond Historical Bond default freq
Rating Class p p* Ratio p*/p p** Ratio p**/p
AAA 0.00015 0.00077 5.2 0.00971 64.7AA 0.0004 0.00185 4.6 0.01362 34.1A 0.00075 0.00322 4.3 0.01721 23.0
BBB 0.0017 0.00659 3.9 0.02393 14.1BB 0.0075 0.02372 3.2 0.04735 6.3B 0.02 0.05438 2.7 0.07995 4.0
CCC 0.08 0.16977 2.1 0.18821 2.4
lambda=0.45 lambda=0.45; k=61-factor transform 2-factor transform
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Shaun Wang, June 2004
Fit 2-factor model to corporate bonds
Bond Rating and Yield Spread
0
200
400
600
800
1,000
1,200
1,400
AAA AA A BBB BB B CCC
Bond Rating
Sp
rea
d (
ba
sis
po
ints
)
Model Fitted Spread
Actual Spread
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Shaun Wang, June 2004
Risk Premium for Corporate Bonds
Use 2-factor Wang transform to fit historical default probability & yield spread by bond rating classes
Compare the fitted parameters for “corporate bond” versus “CAT-bond”
parameters are similar,
“CAT-bond” has lower Student-t degrees-of-freedom,
In 1999, CAT-bond offered more attractive returns for the risk than corporate bonds
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Shaun Wang, June 2004
Cat bond vs. Corporate Bond (before)
Before Sept. 11 of 2001 fund managers were less familiar (or comfortable) with the cat bond asset class.
Fund managers were reluctant to expose themselves to potential career risks, since they would have difficulties in explaining losses from investing in cat bonds, instead of conventional corporate bonds.
At that time, because of investors’ weak appetite for cat bonds, cat bonds issuers had to offer more attractive yields than corporate bonds with comparable default frequency & severity.
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Shaun Wang, June 2004
Cat bond vs. Corporate Bond (after)
During 2002-3, fund managers' interest in cat bond has growth significantly, due to superior performance of the cat bond class. They now complaint about not having enough cat bond issues to feed their risk appetite.
In the same time period, the perceived credit risk of corporate bonds increased, in tandem with the general broader market. Investors began to value more the benefit of low correlation between cat bond and other asset classes.
It has been reported that the yields spreads on cat bonds have tightened while the yields spreads on corporate bonds have widened (cross over) – Polyn April 2003.